Question

Two sinusoidal waves of the same frequency travel in the same direction along a string. If $A_1=3.0cm,A_2=4.0cm,{\varphi }_1=0,$ and ${\varphi }_2=\pi /2rad$, what is the amplitude of the resultant wave?

Answer 1

Let first wave

$y_1=A_1\sin \left(\omega t\right)or,y_1=3\sin \left(\omega t\right)$

and

$y_2=A_2\sin (\omega t+\frac{\pi }{2})=4\sin (\omega t+\frac{\pi }{2})$

resultant wave

$y=y_1+y_2=3\sin \left(\omega t\right)+4\sin (\omega t+\frac{\pi }{2})=3\sin \omega t+4\cos \omega t=5[\sin \omega t\times \frac{3}{5}+\cos \omega t\times \frac{4}{5}]=5\sin (\omega t+\alpha )\{where\cos \alpha =\frac{3}{5}and\sin \alpha =\frac{4}{5}\}$

Hence amplitude of resultant wave is $5cm$